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Showing 151–180 of 1709 reference entries

Mathematics & logicTotal derivative in finite dimensions

Differentiability means a linear approximation with smaller-order error.

DefinitionMathematics & logicDouble integrals on simple regions

A continuous double integral can be evaluated by slicing.

Mathematical statementMathematics & logicGradient, divergence and curl

These coordinate derivatives have different input and output types.

DefinitionMathematics & logicStokes' circulation theorem

Circulation around a boundary equals oriented curl flux.

Mathematical statementMathematics & logicDivergence theorem

Outward boundary flux equals integrated divergence.

Mathematical statementMathematics & logicLocal existence and uniqueness for autonomous ODEs

Local Lipschitz continuity ensures one local trajectory.

Mathematical statementMathematics & logicConstant-coefficient linear ODE systems

The matrix exponential solves a finite linear system.

Mathematical statementMathematics & logicLaplace, heat and wave equation problems

A PDE becomes a solution problem only with a domain and data.

ModelMathematics & logicRandom variables and probability distributions

A distribution assigns probabilities to a random variable's values.

DefinitionMathematics & logicNormal distribution

A normal model is specified by location and positive scale.

DefinitionMathematics & logicBinomial and Poisson count models

Different count mechanisms require different distributions.

DefinitionMathematics & logicWeak law of large numbers

A sample mean concentrates near its mean under stated assumptions.

Mathematical statementMathematics & logicClassical iid central limit theorem

Centered sample averages have an asymptotic normal law.

Mathematical statementMathematics & logicConditional masses and densities

A conditional distribution renormalizes a joint law.

DefinitionMathematics & logicLikelihood and Bayesian updating

Likelihood is a function of a parameter given observed data.

MethodMathematics & logicNull tests, significance and p-values

A p-value is calibrated under a specified null model.

MethodMathematics & logicSampling, randomization and confounding

Selection and treatment assignment answer different questions.

MethodMathematics & logicPermutation inference under exchangeability

Distribution-free procedures still require assumptions.

MethodInformationFinite-alphabet mutual information

Mutual information measures statistical dependence in bits.

DefinitionInformationNear-lossless source coding threshold

Entropy bounds asymptotic compression for an iid source.

Mathematical statementInformationDiscrete memoryless channel capacity

Capacity is the maximal asymptotically reliable rate.

Mathematical statementInformationHamming distance and error correction

Separated codewords permit bounded symbol-error recovery.

Mathematical statementInformationInternet protocol layers

Protocol layers divide communication responsibilities.

DefinitionInformationTCP reliable byte-stream transport

TCP orders bytes and retransmits detected losses.

MethodInformationEncryption, hashes and message authentication

Cryptographic primitives serve distinct security properties.

DefinitionInformationPublic-key digital signatures

A signing key and a verification key have different roles.

DefinitionInformationComputable reductions and undecidability

Reduction direction determines which impossibility transfers.

Mathematical statementInformationDecision classes P and NP

Polynomial solving and polynomial verification differ.

DefinitionInformationNP-hardness and NP-completeness

NP-completeness combines verification and reduction hardness.

DefinitionInformationBounded-error randomized computation

Random choices permit a stated probability of error.

Definition
Mathematical statementSource checked

Hamming distance and error correction

Information

Separated codewords permit bounded symbol-error recovery.

Key knowledge

Hamming distance counts differing coordinates of equal-length words. A binary block code with minimum distance 2t+1 can correct up to t bit substitutions by nearest-codeword decoding and detect up to 2t substitutions by codeword membership.

Conditions and scope

  • t is a nonnegative integer and codewords have one fixed length.
  • Detection and correction are different decoding tasks; insertions and deletions are excluded.
Sources · 1

References

Sources retain their recorded scope. A source link alone does not establish verification of every statement.

  1. Abstract Algebra, 16.1 Error-Detecting and Correcting CodesThomas W. JudsonCourse notesopenmathbooks.orgRead source

Evidence and review

Hamming distance, block-code definitions and Theorem 16.13, odd minimum-distance correction/detection bound.

2026-10-10 · Odd-distance Theorem 16.13 and Hamming metric.

Related reference entries

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Showing 1–4 of 4 relationships

Scientific relationships · 4 / 4

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  1. Uses definition
    Relationship sources · 1

    Linked statements

    • Hamming distance counts differing coordinates of equal-length words. A binary block code with minimum distance 2t+1 can correct up to t bit substitutions by nearest-codeword decoding and detect up to 2t substitutions by codeword membership.
  2. Uses definition
    Relationship sources · 1

    Linked statements

    • Over F₂ take H=[A|Iₘ] and G=[Iₙ₋ₘ;A]. Encoding message x as c=Gx is injective, HG=0, and the code is exactly ker H. The first n−m coordinates hold information and the final m supply parity checks.
  3. Uses definition
    Relationship sources · 1

    Linked statements

    • For received r=c+e with Hc=0, syndrome Hr=He. If H has distinct nonzero columns, a single-bit error's syndrome identifies its position; flip that bit. A zero syndrome indicates a codeword, and this rule uniquely corrects every error of weight at most one.
  4. Related to
    Relationship sources · 1

    Linked statements

    • For a binary symmetric channel with crossover probability 0<p<1/2 and fixed target block error 0<epsilon<1, the best rate has normal approximation C−sqrt(V/n) Q-inverse(epsilon)+O(log(n)/n). Here C=1−h2(p) and V=p(1−p)[log2((1−p)/p)]². Capacity is the leading asymptotic limit; finite length, decoder cost and the remainder matter for actual reliability.

    A realizable code/decoder may operate below the best finite-block rate.

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